Speaking from the scene, Health Minister Wael Abou Faour said more than 200 people had been wounded, many of them in serious condition.
The blasts appeared to mark a return to a campaign of attacks that targeted the group's strongholds between 2013 and 2014, ostensibly in revenge for its military support for Syrian President Bashar al-Assad.
The blasts hit a narrow shopping street in the Burj al- Barajneh neighbourhood that is also home to a street market.
"Soldiers of the Caliphate" detonated explosives planted on a motorbike in an area frequented by Shiites, using a derogatory term to refer to the sect, the statement said.
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"After the apostates gathered in the area, one of the knights of martyrdom detonated his explosive belt in the midst of them," the statement added.
The claim could not be independently verified but it followed the usual format of IS claims of responsibility and was circulated on jihadist online accounts.
The army said the attacks were carried out by two suicide bombers and that the body of a third, who had failed to detonate his explosive device, had been found at the scene of the second blast.
A string of them targeted areas where the group is popular throughout 2013 and 2014.